Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
408 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D131 |
| commutator, socle, prime radical, cosocle | C23×C131 | C24 |
| Fitting | #383×C131 | C2 |
| hypercenter | #383 | D131 |
| hypocenter | C131 | #2863 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #383 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. C42×C131 | Q8 | semidirect |
| 65. C131 | #2863 | semidirect |
| 100. C42 | Dic262 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22553216 | 2 |
| #22553209 | 2 |
| #22553199 | 2 |
| #22553195 | 2 |
| #22553180 | 2 |
| C42×Dic131 | 2 |
| #383×C131 | 2 |
| #2863 | 131 |